Properties of some nonlinear partial dynamic equations on time scales

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1 Malaya Journal of Matematik 4)03) 9 Properties of some nonlinear partial dynamic equations on time scales Deepak B. Pachpatte a, a Department of Mathematics, Dr. Babasaheb Ambedekar Marathwada University, Aurangabad, Maharashtra-43004, India. Abstract The aim of the present paper is to study some basic qualitative properties of solutions of certain nonlinear partial dynamic equations on time scales. The tools employed are application of Banach fixed point theorem and a variant of certain fundamental integral inequality with explicit estimates on time scales. Keywords: Dynamic equations, time scales, qualitative properties, inequalities with explicit estimates. 00 MSC: 6E70, 34N05. c 0 MJM. All rights reserved. Introduction During the past few years many authors have obtained the time scale analogue of well known partial dynamic equations see, 6, 0, ]. Recently in, 3, 9] authors have obtained inequalities on two independent variables on time scales. In the present paper we establish some basic qualitative properties of solutions of certain partial dynamic equation on time scales. We use certain fundamental integral inequalities with explicit estimates to establish our results. We assume understanding of time scales and its notation. Excellent information about introduction to time scales can be found in 4, 5]. In what follows R denotes the set of real numbers, Z the set of integers and T denotes arbitrary time scales. Let C rd be the set of all rd continuous functions. We assume T and T are two time scales and Ω = T T. The delta partial derivative of a function zx, y) for x, y) Ω with respect to x, y and xy are denoted by z x, y), z x, y) and z x, y) = z x, y) with the given boundary conditions. with the given initial boundary conditions u x, y) = f x, y, ux, y), u x, y) ).) u x, ) = αx), u, y) = β y), u, ) = 0,.) for x, y) Ω, where f C rd Ω R n R n, R ), α, β R, R n ). In this paper we study the existence, uniqueness and other properties of the solutions of.)-.) under some suitable conditions on the functions involved in.)-.). The main tool employed is based on application of Banach fixed point theorem 8] coupled with Bielecki-type norm 7] and suitable integral inequality with explicit estimate. Preliminaries and Basic Inequality For a function ux, y) and its delta derivative u x, y) in C rd Ω, R n ) we denote by ux, y) W = u x, y) u x, y). For t, s) Ω the notation at, s) = Obt, s)) then there exists a constant q > 0 such that Corresponding author. addresses: Deepak B. Pachpatte)

2 Deepak B. Pachpatte / Properties of some... at,s) bt,s) q right-hand neighborhood. Let G be the space whose functions φ x, y), φ x, y) ) W which are rd-continuous for x, y) Ω and satisfy the condition for x, y) Ω, where λ > 0 is a constant. In space G we define the norm φ G = φ x, y) W = O e λ x, y)),.) sup φ x, y) W e λ x, y)]..) x,y) Ω It is easy to see that G with norm defined in.) is a Banach space. The condition.) implies that there is a constant N 0 such that φ x, y) W Ne λ x, y)..3) Using the fact that in.) we observe that φ x, y) W N..4) By a solution of.).) we means a function ux, y) C rd Ω, R n ) which satisfies the equation.).). It is easy to see that solution ux, y) of.).) satisfies the following equation u x, y) = α x) f x, t, ux, t), u x, t) ) t,.5) and u x, y) = α x) β y) f s, t, us, t), u s, t) ) t s,.6) for x, y) Ω. We require the following integral inequality. Lemma.. Let u, a, b C rd Ω, R n ) u x, y) c ax, t)ux, t) t y b s, t)u s, t) t s,.7) for x, y) Ω, then for x, y) Ω, where for x, y) Ω. u x, y) ch x, y) e y bs,t)hs,t) t x, ),.8) Hx, y) = e a y, ),.9) Proof. Define a function nx, y) by then.7) can be restated as nx, y) = c u x, y) nx, y) b s, t)u s, t) t s,.0) ax, t)ux, t) t..) Clearly nx, y) is nonnegative for x, y) Ω and nondecreasing for x. Treating.) as a one-dimensional integral inequality and a suitable application of inequality given in Theorem 3. ] yields u x, y) nx, y)hx, y),.)

3 Deepak B. Pachpatte / Properties of some... 3 where Hx, y) is given by.9). From.0) and.) we have nx, y) c Now a suitable application of Theorem. 9] to.3) yields nx, y) ce y b s, t)hs, t)n s, t) t s..3) bs,t)hs,t) t Using.4) into.) we get the required inequality in.8). x, )..4) 3 Main results Our main results are given in the following theorems. Theorem 3.. Suppose that i) the function f in.) satisfies the condition f x, y, u, v) f x, y, u, v) p x, y) u u v v ], 3.) where p C rd Ω, R n ), ii) for λ as in.) a) there exists a nonnegative constant γ such that γ < and px, t)e λ x, t) t b) there exists a nonnegative constant η such that α x) β y) α x) y p s, t)e λ s, t) t s γe λ x, y), 3.) f x, t, 0, 0) t f s, t, 0, 0) t s ηe λ x, y), 3.3) for x, y) Ω where α, β are the functions given in.). Then.).) has a unique solution in G. Proof. Let ux, y) G and define operator S by Su) x, y) = α x) β y) f s, t, u s, t), u s, t) ) t s. 3.4) Delta differentiating both sides of 3.4) with respect to x we get Su) x, y) = α x) y f x, t, u x, t), u x, t) ) t. 3.5) First we show that Su maps G into itself. Evidently Su) is rd-continuous. We verify that.) is fulfilled.

4 4 Deepak B. Pachpatte / Properties of some... From 3.4) and 3.5) using the hypotheses and.3) we have Su) x, y) Su) x, y) α x) β y) α x) f s, t, 0, 0) t s f x, t, 0, 0) t ηe λ x, y) f s, t, us, t), u s, t)) f s, t, 0, 0) t s f x, t, us, t), u x, t)) f x, t, 0, 0) t p x, t) e λ x, t) ux, t) G e λ x, t) t p s, t)e λ s, t) u s, t) G e λ s, t) t s ηe λ x, y) u s p x, t) e λ x, t) t p s, t)e λ s, t) t s η Nγ] e λ x, y). 3.6) From 3.6) it follows that Su G. This proves S maps G into itself. Now we verify S is a contraction map. Let ux, y), vx, y) G. From 3.4) and 3.5) and using hypotheses we have Su) x, y) Sv) x, y) Su) x, y) Sv) x, y) f s, t, u s, t), u s, t) ) f s, t, v s, t), v s, t) ) t s f x, t, u x, t), u x, t) ) f x, t, v x, t), v x, t) ) t p x, t) e λ x, t) u x, t) v x, t) G e λ x, t) t p s, t)e λ s, t) u s, t) v s, t) G e λ s, t) t s u v G γe λ x, y). 3.7) Consequently from 3.7) we have Su Sv G γ u v G. Since γ <, it follows from Banach fixed point theorem that S has a unique fixed point in G. The fixed point of S is however solution of.).). The proof is complete. Now we give theorem concerning the uniqueness of solutions of.).) without existence. Theorem 3.. Assume that the function f in.) satisfies the condition 3.). Then.).) has at most one solution on Ω.

5 Deepak B. Pachpatte / Properties of some... 5 Proof. Let u x, y) and u x, y) be two solutions of.).). Then by hypotheses we have u x, y) u x, y) u x, y) u x, y) f s, t, u s, t), u s, t) ) f s, t, u s, t), u s, t) ) t s f x, t, u x, t), u x, t) ) f x, t, u x, t), u x, t) ) t p x, t) p s, t) u x, t) u x, t) u x, t) u x, t) ] t u s, t) u s, t) u s, t) u s, t) ] t s. 3.8) Now an application of Lemma. with ax, y) = bx, y) = px, y) and c = 0) to 3.8) yields. u x, y) u x, y) u x, y) u x, y) 0, 3.9) which implies u x, y) = u x, y) for x, y) Ω. Thus there is at most one solution of.).) on Ω. 4 Boundedness and continuous dependence In this section we study the boundedness of solution of.).) and the continuous dependence of solutions of equation.) on the given initial boundary values, the function f involved therein and also the continuous dependence of solutions of equations of the.) on parameters. The following theorem concerning the estimate on the solution.).) holds. Theorem 4.. Assume that α x) β y) α x) k, 4.) f x, y, u, v) r x, y) u v ], 4.) where k 0 is a constant r C rd Ω, R n ). If ux, y), x, y) Ω is any solution of.).) then u x, y) u x, y) kq x, y) e y rs,t)qs,t) t x, ), 4.3) for x, y) Ω, where for x, y) Ω. q x, y) = e r y, ), 4.4) Proof. Using the fact that ux, y) is a solution of.).) and conditions 4.) 4.) we have u x, y) u x, y) α x) β y) α x) f s, t, u s, t), u s, t) ) t s k r x, t) u x, t) u x, t) ] t f x, t, u x, t), u x, t) ) t r s, t) u s, t) u s, t) ] t s. 4.5) Now a suitable application of Lemma. to 4.5) yields 4.3).

6 6 Deepak B. Pachpatte / Properties of some... Remark 4.. The estimates obtained in 4.3) yields not only the bound on the solution ux, y) of.).) but also the bound on u x, y). If the estimate on the right side in 4.3) is bounded then the solution of ux, y) of.).) and also u x, y) are bounded on Ω. Our next result deals with the continuous dependence of solutions of equation.) and given initial boundary conditions. Theorem 4.. Let u x, y) and u x, y) be the solution of equation.) and given initial boundary conditions u x, ) = α x), u, y) = β y), u, ) = 0, 4.6) u x, ) = α x), u, y) = β y), u, ) = 0, 4.7) respectively where α, α, β, β R, R n ). Suppose that the function f in equation.) satisfies the condition 3.) and for x, y) Ω, where d 0 is a constant. Then α x) β y) α x) β y) α x) α x) d, 4.8) u x, y) u x, y) u x, y) u x, y) dq x, y) e y ps,t)qs,t) t s x, ), 4.9) for x, y) Ω, where qx, y) is defined by the right hand side of 4.4) replacing rx, y) by px, y) for x, y) Ω. Proof. From the hypotheses it is easy to observe that u x, y) u x, y) u x, y) u x, y) α x) β y) α x) β y) α x) α x) f s, t, u s, t), u s, t) ) t s f s, t, u s, t), u s, t)) f x, t, u x, t), u x, t) ) t f x, t, u x, t), u x, t)) ] d p x, t) u x, t) u x, t) u x, t) u x, t) t p s, t) u s, t) u s, t) u s, t) u ] s, t) t s. 4.0) Now a suitable application of Lemma. to 4.0) yields the bound in 4.9), which shows dependency on solution of equation.) on given initial boundary values. Consider the initial boundary value problem.)-.) and the corresponding initial boundary value problem v x, y) = F x, y, v x, y), v x, y) ), 4.) v x, ) = α x), v, y) = β y), v, ) = 0, 4.) for x, y) Ω where v C rd Ω, R ), α, β R, R n ) and F C rd Ω R n R n, R n ). The next theorem deals with continuous dependence solutions of initial boundary value problem.)-.) on the functions involved therein.

7 Deepak B. Pachpatte / Properties of some... 7 Theorem 4.3. Assume that the function f in equation.) satisfies the condition 3.). x, y) Ω be a solution of initial boundary value problem 4.) 4.) and α x) β y) α x) β y) α x) α x) f x, t, v x, t), v x, t) ) F x, t, v x, t), v x, t) ) t Let vx, y) for f s, t, v s, t), v s, t) ) F s, t, v s, t), v s, t) ) t s ɛ, 4.3) for x, y) Ω where α, β, F are functions involved in initial boundary value problem.).) and initial boundary value problem 4.) 4.) and ɛ 0 is a constant. Then the solution ux, y) on initial boundary value problem.).) depends continuously on the functions involved therein. Proof. Since ux, y) and vx, y) are solutions of initial boundary value problem.)-.) and initial boundary value problem 4.) 4.) and the conditions 3.), 4.3) we get u x, y) vx, y) u x, y) v x, y) α x) β y) α x) β y) α x) α x) f s, t, u s, t), u s, t) ) f s, t, v s, t), v s, t) ) t s f s, t, v s, t), v s, t) ) F s, t, v s, t), v s, t) ) t s f x, t, u x, t), u x, t) ) f x, t, v x, t), v x, t) ) t f x, t, v x, t), v x, t) ) F x, t, v x, t), v x, t) ) t ε p x, t) u x, t) v x, t) u x, t) v x, t) ] t p x, t) u s, t) v x, t) u s, t) v s, t) ] t s. 4.4) Now a suitable application of Lemma. to 4.4) yields u x, y) vx, y) u x, y) v x, y) εq x, y) e y ps,t)qs,t) x, ). 4.5) Now we consider following equation with the given initial boundary conditions z x, y) = f x, t, z x, t), z x, t), µ ), 4.6) z x, y) = f x, t, z x, t), z x, t), µ 0 ), 4.7) z x, ) = τ x), z, y) = ψ y), z, ) = 0, 4.8) where z C rd Ω, R ), τ, ψ R, R n ), f C rd Ω R n R n R, R n ) and µ, µ 0 are real parameters. Finally, we present following theorem which deals with continuous dependency of solutions of initial boundary value problem 4.6) 4.8) and initial boundary value problem 4.7) 4.8) on parameters.

8 8 Deepak B. Pachpatte / Properties of some... Theorem 4.4. Assume that the function f in 4.6) and 4.7) satisfy the conditions for x, y) Ω where n, m C rd Ω, R ) and f x, y, u, v, µ) f x, y, u, v, µ) h x, y) u u v v ], 4.9) f x, y, u, v, µ) f x, y, u, v, µ 0 ) m x, y) µ µ 0, 4.0) m x, y) t m s, t) t s δ, 4.) where δ 0 is a constant. Let z x, y) and z x, y) be the solutions of initial boundary value problem 4.6) 4.8) and initial boundary value problem 4.7) 4.8) respectively. Then z x, y) z x, y) z x, y) z x, y) µ µ 0 δq x, y) e y x, ), 4.) hs,t)qs,t) t for x, y Ω, where for x, y Ω. Q x, y) = h x, t) t, 4.3) Proof. Since z x, y) and z x, y) be the solutions of initial boundary value problem 4.6) 4.8) and and initial boundary value problem 4.7) 4.8) and conditions 4.9) 4.) we have z x, y) z x, y) z x, y) z x, y) f s, t, z s, t), z s, t), µ ) f s, t, z s, t), z s, t), µ ) t s f s, t, z s, t), z s, t), µ ) f s, t, z s, t), z s, t), µ 0 ) t s ) f x, t, z x, t), z x, t), µ f s, t, z x, t), z x, t), µ ) t ) f x, t, z x, t), z x, t), µ f ) s, t, z x, t), z x, t), µ 0 t h s, t) h x, t) z s, t) z s, t) z s, t) z s, t) ] t s m s, t) µ µ 0 ] t s z x, t) z x, t) z x, t) z x, t) ] m x, t) µ µ 0 ] t µ µ 0 δ h x, t) z x, t) z x, t) z x, t) z x, t) ] t

9 Deepak B. Pachpatte / Properties of some... 9 h s, t) z s, t) z s, t) z s, t) z s, t) ] t s. 4.4) Now a suitable application of Lemma to 4.4) yields 4.) which shows the dependency of solutions of initial boundary value problem 4.6) 4.7) and initial boundary value problem 4.7) 4.8) on parameters. Remark 4.. The results obtained above can be very easily extended to the following partial dynamic equation on time scales u x, y) = f x, y, ux, y), u x, y) ).) with the initial boundary conditions in.) by modifying suitably the inequality given in Lemma. References ] C. D. Ahlbrandt, Ch. Morian, Partial differential Equations equations on time scales, J. Comput. Appl. Math., 400), ] D. R. Anderson, Dynamic Double integral inequalities in two independent variables on time scales, J. Math. Inequal., 008), ] D. R. Anderson, Nonlinear Dynamic integral inequalities in two independent variables on time scale, Adv. Dyn. Syst. Appl., 3008), -3. 4] M. Bohner and A. Peterson, Dynamic equations on time scales, Birkhauser, Boston/Berlin, 00. 5] M. Bohner and A. Peterson, Advances in Dynamic equations on time scales, Birkhauser, Boston/Berlin, ] M. Bohner and G. Guseinov, Partial Differentiation on time scales, Dynamic Systems and Applications, 3004), ] A. Bielecki, Un remarque sur I application de la methode de Banach Caccipoli-Tikhnov dans la theorie de I equation s=fx,y,z,p,q), Bull. Acad. Polon. Sci Ser. Sci Math. Phys. Astr., 4956), ] C. Corduneanu, Integral Equations and Applications, Cambridge University Press,99. 9] R A C Ferreira, Some linear and nonlinear integral inequalities on time scales on two independent variables, Nonlinear Dynamics and Systems Theory, 9)009), ] J. Hoffacker, Basic partial dynamic equations on time scales, J. Difference Equ. Appl. Math., 800), ] B. Jackson, Partial dynamic Equations equations on time scales, J. Comput. Appl. Math., 86006), ] D. B. Pachpatte, Explicit estimates on integral inequalities on time scales, J. Inequal. Pure Appl. Math., 6)005), Art 43. Received: May, 0; Accepted: June 08, 03 UNIVERSITY PRESS